[Paper Review] Using D-operators to construct orthogonal polynomials satisfying higher order q-difference equations

R. Álvarez-Nodarse, Antonio J. Durán|Deposito de Investigacion Universidad de Sevilla (University of Seville)|Sep 12, 2013

Mathematical functions and polynomials17 references3 citations

TL;DR

This paper introduces a novel method using D-operators to construct q-Krall polynomials—orthogonal polynomials that are also eigenfunctions of higher-order q-difference operators—by modifying q-Meixner and q-Laguerre polynomials via linear combinations with coefficient sequences (βₙ). The key contribution is a systematic construction of such polynomials with orthogonality measures obtained through Christoffel transforms, yielding eigenfunctions of q-difference operators of order 2k+2 or 2α+2.

ABSTRACT

Let $(p_n)_n$ be either the $q$-Meixner or the $q$-Laguerre polynomials. We form a new sequence of polynomials $(q_n)_n$ by considering a linear combination of two consecutive $p_n$: $q_n=p_n+β_np_{n-1}$, $β_n\in \RR$. Using the concept of $\D$-operator, we generate sequences $(β_n)_n$ for which the polynomials $(q_n)_n$ are orthogonal with respect to a measure and common eigenfunctions of a higher order $q$-difference operator.

Motivation & Objective

• To extend the theory of Krall polynomials to the q-difference setting by constructing q-Krall polynomials from classical q-orthogonal polynomials.
• To address the open problem of generating orthogonal polynomials that are common eigenfunctions of higher-order q-difference operators, a q-analogue of Krall's classical problem.
• To develop a systematic method using D-operators to generate sequences (βₙ) such that modified polynomials qₙ = pₙ + βₙpₙ₋₁ are orthogonal and eigenfunctions of higher-order q-difference operators.
• To characterize the resulting orthogonality measures as Christoffel transforms of the original q-Meixner and q-Laguerre measures, incorporating polynomial factors and Dirac deltas.

Proposed method

• Define a D-operator associated with a sequence of q-classical polynomials (q-Meixner or q-Laguerre) and the algebra A_q of finite-order q-difference operators.
• Use the D-operator to generate coefficient sequences (βₙ) such that the modified polynomials qₙ = pₙ + βₙpₙ₋₁ (with p₋₁ = 0) are orthogonal with respect to a transformed measure.
• Construct the orthogonality measure as a Christoffel transform: μ = w(x)ρ, where ρ is the original q-Meixner or q-Laguerre measure and w(x) is a polynomial factor.
• For q-Meixner, three distinct D-operators yield three different (βₙ) sequences, each producing orthogonal qₙ with respect to a distinct measure.
• For q-Laguerre, use the Al-Salam-Carlitz polynomials and a moment functional involving a Dirac delta at 0 to define βₙ via γₙ = vₖ¹ᐟqᵃ;𝑞(qⁿ) and βₙ = γₙ₊₁/γₙ.
• Prove that the resulting polynomials qₙ are eigenfunctions of a q-difference operator of order 2k+2 (for q-Meixner) or 2α+2 (for q-Laguerre) by verifying the eigenvalue equation using the D-operator structure.

Experimental results

Research questions

• RQ1 Can D-operators be used to systematically generate q-Krall polynomials that are eigenfunctions of higher-order q-difference operators?
• RQ2 What conditions on the coefficient sequences (βₙ) ensure that the modified polynomials qₙ = pₙ + βₙpₙ₋₁ remain orthogonal with respect to a transformed measure?
• RQ3 How do the resulting orthogonality measures relate to the original q-Meixner and q-Laguerre measures, and what role do polynomial factors and Dirac deltas play?
• RQ4 What is the precise order of the q-difference operator for which the constructed qₙ are eigenfunctions, and how does it depend on parameters like k or α?
• RQ5 Can the method be generalized to other q-classical families, and what are the structural constraints on the coefficient sequences (βₙ) for such constructions?

Key findings

• For q-Meixner polynomials, three distinct D-operators generate three different sequences (βₙ), each producing a family of orthogonal polynomials (qₙ) that are eigenfunctions of a q-difference operator of order 2k+2.
• The orthogonality measures for the qₙ are Christoffel transforms of the q-Meixner measure, specifically μ = w(x)ρ, where w(x) is a polynomial of degree k.
• For q-Laguerre polynomials with α a positive integer, the construction yields orthogonal polynomials (qₙ) that are eigenfunctions of a (2α+2)-th order q-difference operator.
• The coefficient sequence βₙ is explicitly defined via βₙ = -γₙ₊₁ / [(1 - q^{α+n})γₙ], where γₙ = 1 + M(q^{α+1};q)ₙ / (q;q)ₙ, ensuring orthogonality with respect to a measure μ = Mδ₀ + ρ_{α-1}^q.
• The method confirms that the q-Krall polynomials constructed are eigenfunctions of higher-order q-difference operators, generalizing earlier results by Grünbaum and Haine.
• Conjectures suggest that for any finite set F of positive integers, the measure ∏_{f∈F}(1 + xq^f)ρ_α^q yields q-Krall polynomials eigenfunctions of a q-difference operator of order 2∑f∈F f - n_F(n_F - 1) + 2.